between any two consecutive squares, there exist at least two primes. the proof is for the first prime; the existence of the second prime can be proven similarly
One way I can prove the two primes theorem is:
let p be the greatest prime
p | p^2 and p | p(p+1)
p(p+1) - p^2 = p, therefore there are p integers greater than p^2 and less than or equal to p(p+1)
however, p^2 is an additional data point, so there are actually p+1 integers between p^2 and...
probability generally deals with a continuous distribution, but logic is typically discrete. Furthermore, probability has as its range all real numbers between 0 and 1, and logic is at its most complex form countably infinite.
however my research deals with any prime number of states (i.e. 5 or...
I agree, but I wanted to begin with a basic example that defines the concept of logical states. A more involved example is the algebraic extension of prime-state logic; however, it is still useful to restrict my range to 2 states. I can define an operator on the domain of statements as a...
that is why I used quotes, they are not really points that terminate the line. they are more like the boundary of infinity; two coincident lines can "grow" at different rates, and the line that grows fastest will enclose the other line. The enclosed line would have endpoints within the outer...
well it is a kind of obscure example, but I am using more general definitions of statements and predicates. a statement is any quantity, whether it is defined or not, and what I called predicates are more like outcomes, so I will use the latter term to describe them
in my example, each cell is...
from a topological point of view, both structures are infinite; without beginning or end. ie the "endpoints" of a line can coincide at infinity, and thus form a closed loop topologically equivalent to a circle. Furthermore, any segment of the line WILL contain the "center" of the line. I...
that is what I meant by a differential scale
also, you are right, a tangent can intersect a cubic curve in more than one point, and that is just one case...
at a small enough scale, a circle is identical to a line; i am referring to a differential scale... however, a tangent will never intersect any curve more than once.
I think you are referring to the mean? the median is defined as the central data point via the exclusion of extrema.
where do the percentages come from?
Unfortunately, I am not done with the proof, as I now have two sub-cases to prove.
Logical states are basically a numeric mapping between predicates and statements. In this case I am using a more general definition of "predicate" than is typically used in predicate logic; they (predicates)...
My theory of algebraic logic is that a logical vector can be transformed into an algebraic vector by multiplying the logical vector by the inverse of a transformation matrix. this theory only works for structures with a prime number of states
another part of the theorem is that there exist primes p and q such that n^2 < p <= n(n+1) < q < (n+1)^2 where n is an integer greater than or equal to 1.